Math Type

Tuesday, February 25, 2014

Book Review: The Math Book by Clifford Pickover

The Math Book by Clifford Pickover was a good read.  The book goes through history and talks about the relevant mathematical findings.  Pickover starts from c. 150 million years B.C. and works his way up to 2007.  Pickover, in his introduction, states that he realizes that some of the discoveries not mentioned may be more important than the ones that he states, but this is his own opinion on the most significant findings and the ones he enjoyed learning about the most.

The book has a good structure.  Each new page is a new "section" and is completely independent from the previous or next page.  Each page also has an accompanying picture that, in some ways, tries to help the reader in their understanding of the subject.  Pickover, in each entry, tries to give some background about the subject and who discovered it and also tries to explain some what the discovery did or is doing for math now.  Each section is short and easy to read, which makes the book a fast read and not too time intensive to read through.

The book did some things well.  The book did a good job going over many topics (249 in total) relatively fast and briefly.  This book gives the reader a good sense of what has been done in math since c. 150 million years B.C.  The book also does a good job trying to link together many of the subjects talked about.  Each page has bolded words that refer the reader to other passages with similar context or to help the reader with an understanding on how the discovery was made (usually due to something earlier in the book).  At the end of each page, you can use the references to find other similar works if reading this book for a leisure read and wanting to gain more knowledge on a specific topic.  Pickover does do a good job of explaining who discovered each mathematical finding too.  He gives a very brief history on the person (i.e. who they are, their nationality, their race/religion, etc.) so that the reader can gain a respect for the breadth of math.  I believe that Pickover does this because he wants the reader to understand that math is discovered everywhere and by all types of people, yet sometimes he can bog down the reader with the details.  The most interesting part of this book, which was sometimes the most frustrating too, was the "paradoxes" and "problems" that Pickover places in the book.  I had an easy time understanding some of the "paradoxes" and/or "problems," but other times, like other sections, I just had no idea what was going on and so I would just skim over the reading and not understand it.  The "paradoxes" and "problems" that I did understand were very interesting though.  I found myself waiting to read the answer until giving myself sufficient time to try and find the answer (or have a guess) myself.  It made the book more fun to read and gave a break from the strictly math sections of the book.

The book also has many things that I would change.  There are frequently (more so toward the end of the book) pages and sections that I would leave out entirely.  For the standard student, some sections are either WAY over their head, way to rudimentary, or just plain boring.  The book would therefore go into way too much detail or just not enough at all.  I found that some sections I would finish the page description and leave with a "what?" or "I don't understand this at all" feel.  After talking to some other students reading the book, they felt the same way.  On the other hand, the pages that were just plain simple, I felt like the discovery was important, but the details not so much because they are self explanatory (i.e. a least squares line he goes into a two paragraph description of what it is).  Another thing that I would change about the book is the amount of background given for each section.  Sometimes the amount of background given for a particular section would overtake the math of the section and cause me to miss the overall reason for the section in the book.  The background, while I understand is sometimes necessary, would just be too in-depth.  A more brief background would have benefited me more and then if I was interested in the subject I would look up more.  The last thing that I did not like about the book was the amount of quotes given from other mathematicians about a particular work.  These added nothing to the sections.  Sometimes it seemed that Pickover would add these quotes just because he realized that a finding was important, but didn't understand if fully and so he would take up space on the page with these quotes.  I found that by the end of the book, almost any time something was quoted from some other mathematician I would completely skip the quote because it added nothing to my understanding and was a waste of my time to read it.

While it may seem that I had a lot of complaints about the book, I did highly enjoy reading it.  I found that as I was reading about things I would be able to link them to other things I have learned in math.  I also enjoyed learning about the history of math from another perspective.  If you can get past the critics I have on the book, I would suggest reading this book, as it was not a hard read and had some very interesting topics.

Monday, February 17, 2014

History of Math: Fibonacci Sequence

Fibonacci has done many things that have advanced mathematics.  Most of his accomplishments came around 1200 A.D. since he had lived from c.1170 to c.1250.  One of his most well known works is the Fibonacci Sequence.  This sequence involves adding the latest two terms in the sequence in order to achieve the new term, written as F(n+1)=F(n)+F(n-1).  This mathematical equation is not too hard to derive, especially given that the first two terms are defined to be 1 and 1 or 0 and 1 depending who you ask.  There are other applications that add to this mathematical equation as well.

Using the Fibonacci Sequence, the Golden Spiral can be made.  Using the Fibonacci numbers as guidelines, you can make squares that continually grow.  Then, connecting corner to corner with an arc, a spiral can be made.  The reason that this spiral has a special name, the Golden Spiral, is because in order to make this spiral, the dimensions are based off of the golden rectangle.

However, how would you answer the problem, how do the Fibonacci Sequence and Pascal's Triangle relate?  Give an example such as 89 to help with understanding?  After some research, you may find that the Fibonacci Sequence is made up of the additions of Pascal's Triangle's angles.  Each line, always drawn parallel to the last, will make up the next number in the Fibonacci Sequence.  So with Pascal's Triangle and the Fibonacci Sequence in hand, errors can be checked for on both.  A good example is the number 89.  89 is the 11th number in the Fibonacci Sequence.  This number is attained by adding 55 and 34 from the Fibonacci Sequence.  Using Pascal's Triangle, 89 is attained by adding 1, 9, 28, 35, 15, and 1 (if you want to see more check here

Sunday, February 9, 2014

Nature of Mathematics: Connections between Algebra and Geometry

Algebra and Geometry both are derived from our understanding of numbers.  Without one, the other would never have existed.  The two mathematical studies evolved concurrently, because advances in Geometry would merit advances in Algebra and the same was true the opposite direction.

Geometry was the beginnings of Algebra.  Through Geometry, mathematicians could visualize different problems that they had been working on.  A good example is learning about different algebraic equations through the usage of squares and other geometric objects.

After many years, advances in Algebra started to lead the way to advances in Geometry.  Through different equations and algebraic problems, different geometric figures could be constructed or at least found to at least exist.

Algebra and Geometry do not cause the discovery of the biggest ideas in either category today.  Algebra also did not cause the discovery of the largest finding in Geometry.  Through the continued practice of Geometric figures, non-Euclidean geometry was found by Gauss, Boylai, and Lobachevsky, all independently of one another.  They found a contradictions to Playfair's Postulate.

While Algebra did not cause the biggest discovery in Geometry, I believe that Geometry did cause the discovery of the biggest Algebra discovery.  By looking at triangles and other geometric figures (mostly squares), the discovery of Pythagorean's Theorem was made.